gibson:juliablog:kuramoto_sivashinksy
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gibson:juliablog:kuramoto_sivashinksy [2017/07/09 06:04] gibson |
gibson:juliablog:kuramoto_sivashinksy [2017/07/09 07:30] (current) gibson |
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| ====== Benchmarking a simple PDE solver in Julia and other languages ====== | ====== Benchmarking a simple PDE solver in Julia and other languages ====== | ||
| + | ==== Julia ==== | ||
| Julia is an innovative new programming language that promises to revolutionize scientific computing. In a nutshell, it is | Julia is an innovative new programming language that promises to revolutionize scientific computing. In a nutshell, it is | ||
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| Julia's main innovation is a carefully-designed type system combined with just-in-time compilation. The combination allows high-level user code to be compiled to machine-code on-the-fly. | Julia's main innovation is a carefully-designed type system combined with just-in-time compilation. The combination allows high-level user code to be compiled to machine-code on-the-fly. | ||
| + | ==== The benchmark algorithm ==== | ||
| The benchmark algorithm here is a simple time-integration of the Kuramoto-Sivashniksy equation | The benchmark algorithm here is a simple time-integration of the Kuramoto-Sivashniksy equation | ||
| - | \begin{align*} | + | \begin{eqnarray*} |
| u_t = -u_{xx} - u_{xxxx} - u u_x | u_t = -u_{xx} - u_{xxxx} - u u_x | ||
| - | \end{align*} | + | \end{eqnarray*} |
| - | on a 1d periodic domain $[0, L_x]$, with $x$ space and $t$ time, and where subscripts indicate differentiation. | + | on a 1d periodic domain $[0, L_x]$, with $x$ space and $t$ time, and where subscripts indicate differentiation. The algorithm uses a Fourier decomposition in space and 2nd-order Crank-Nicolson, Adams-Bashforth semi-implicit finite-differencing in time, with collocation computation of the nonlinear term $u u_x$. I implemented the same algorithm in Python, Matlab, C++, and in two forms in Julia. The codes and a detailed description of the algorithm is given below. |
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| + | ==== The results ==== | ||
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| + | **The left plot** shows execution time of 3200 time steps of the algorithm as a function of $N_x$, the number of gridpoints in the Fourier decomposition. The dominant cost of the algorithm should be the FFTs, which should scale as $N_x \log N_x$. All the codes use the same FFTW libraries, so ideally, they should all collapse onto the same $N_x \log N_x$ line as linear and fixed-size overheads costs decrease relative to that. | ||
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| + | {{:gibson:juliablog:cputime.png?400|}} {{:gibson:juliablog:timeloc.png?400|}} | ||
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| + | The right plot shows | ||
gibson/juliablog/kuramoto_sivashinksy.1499605496.txt.gz · Last modified: 2017/07/09 06:04 by gibson
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